7 ms·
How to Fold a Julia Fractal
- nicholassmith 12y agoI always love reading these, even when I don't understand some of the maths involved.
- lwhi 12y agoThe interactive illustrations are brilliant. They really aid comprehension.
- bsilvereagle 12y agoPhasors finally clicked after watching the first one. Author uses MathBox which he appears to have developed: http://acko.net/blog/making-mathbox/ http://acko.net/blog/making-mathbox/
- Gracana 12y agoI wish my math classes in high school and college had stuff like this. It's amazing how much easier it is to understand something when you can see and interact with it.
- woah 12y agoWow, I didn't even read the article at all, but the background and title of the site are amazing. Some kind of 3d parallax rendering on canvas. As I was examining it, a badge popped up, I had unlocked "Dat Parallax".
- puzzlingcaptcha 12y agoprevious discussion: https://news.ycombinator.com/item?id=6268610 https://news.ycombinator.com/item?id=6268610
- j2kun 12y agoExcellent post! > To understand the effect of c we need to make a Mandelbrot set. In fact, the Mandelbrot set can be defined to be the set of c that make the Julia set connected. This is something new I learned today because I was suspicious about the claims of connectivity (proving fractals are connected is a highly nontrivial task). Now I kinda want to start looking at MathBox for presentations.
- coreymgilmore 12y agoGreat read. Numbers are great, and there is so much hidden between/under them. Also, very nice graphics.
- fjcaetano 12y agoStunning visualisation. Really eased the comprehension of what's being described.
- Gracana 12y agoOuch, this page absolutely destroys my browser (up-to-date Firefox on Windows 7.) 20 seconds of unresponsiveness on load, followed by very choppy scrolling and animation.
- gavinpc 12y agoThe worst part is that it's not the article, but the template that's killing it. I love the visualizations, but I think the author needs to reconsider including the parallax trickery on inner pages. There is enough to look at, and, when it's not busy crashing your browser, this kind of eye candy is a distraction.
- jayflux 12y agoThis crashed my browser for about 10 seconds too, up-to-date Chrome on Windows 7. I agree about the parallax stuff, its great, but when you're sitting there waiting for your browser to come back alive it sort of puts you off visiting the site again
- jonahx 12y agomy new macbook pro got too hot for my lap within about 30 seconds....
- Cowicide 12y agoWeird, the old MacBook Pro with Snow Leopard I'm using with old Safari 5.1.x didn't even rev up the fans or anything. Which OS version and browser are you using?
- anon4 12y agoMine just died. Firefox 29.0.1 on Linux. I had to disable webgl.
- saidajigumi 12y agoFWIW, no problems at all on FF 30, Macbook Pro, OS X 10.9.3. Responsiveness and framerates were good, and no unusually high CPU logged for the entire time I was reading the article.
- deleted 12y ago[deleted]
- pdkl95 12y agoI'm not sure if it was the visual aids (which were outstanding), or simply how the subject was approached and written about, but that article finally gave me an intuitive understanding of "complex numbers" that i've never quite gotten before. Sure, they were nice for some types of coordinate/vector math, but why they were nice always seemed to be something I missed. The warping of the coordinate system that you get when squaring is probably the key. It totally explained Julia sets for me. Now if only I could get the same understanding for the incredible complexity of the Mandelbrot. After seeing[1] what happens down around 1/ 2^1116 (~3E353) - and remember that the set is supposed to be simply-connected - I'm not sure how to begin any kind of real understanding. There is simply too much detail there... [1] http://www.youtube.com/watch?v=PbwaFQ2r2c4 http://www.youtube.com/watch?v=PbwaFQ2r2c4 (the entire video is great, but the true insanity starts about halfway through)
- alphonse23 12y agoI felt the same way. A lot of appreciation goes to the writer of that piece!
- regularfry 12y agoThe visualisations are broken for me, Chromium 34.0.1847.137 Debian 7.5 (268882) and Iceweasel 24.5.0.
- bsilvereagle 12y agoThe visualizations are WebGL dependent. Check http://get.webgl.org/ http://get.webgl.org/ to see if WebGL works for your system.
- regularfry 12y agoYep, WebGL works on my system. Both browsers show the spinning cube fine.
- dvdplm 12y agoAmazing work. Love this.
- mitosis 12y agoHeh: "Achievement unlocked: Refresh Prince of Bel Air".
- roadnottaken 12y agoThis guy is so talented it's actually depressing!
- greggman 12y agoI love the visualization on acko.net but for whatever reason, unlike apparently many of you, I still don't understand the topics he's trying to explain. As just one example on step 18 of the first major visualization it says "And that's actually a remarkable thing, because it means our invented rule has created a square root of −1. It's the number 1∠90∘" WAT?! I'm sure someone skilled at math understands that but as a lamer at math I have no idea how that explains that the rule invented the square root of -1. Nothing before that has mentioned square roots or how this vector + rotation has any relation to them.
- milu 12y agoI had the same reaction. There should be a few words showing that we now have a point which multiplied by itself (rotated by 90° around 0) goes to -1. So this number is the square root of -1.
- qiemem 12y agoThe next slide shows that.
- qiemem 12y agoI think that particular slide was poorly worded. He's actually introduction a consequence of the rules, but not saying it's an obvious consequence. Slide 19 shows how (1∠90∘)2 = -1.
- vitd 12y agoI don't want to disparage the author's hard work, as it was quite beautiful to look at, but as someone who already understands what he's talking about, I felt this was a terrible way to present it. (Like I say, it looked great, but I don't think it would aid a new student learning about complex numbers at all.) The notation he invents is confusing, too. That "1∠90°" just means a vector with length 1 and pointing 90° from the X-axis (which the author later changes to the Y-axis for no apparent reason). In my opinion, polar coordinates (which is what the notation represents) are a fascinating tool that can be very helpful. I use them quite a bit. Conflating that with complex numbers is confusing if you don't already understand both polar coordinates and complex numbers, in my opinion.
- 8iterations 12y agoCan you place links at each of your figures so that we can directly link to those of interest? This is great stuff!
- hcarvalhoalves 12y agoThat's blogging on a whole new level.
- Jemaclus 12y agoSome of the math was over my head, but the visuals on this are amazing.
- KhalilK 12y agoAnyone has an idea how to create such smooth animations?
- pornel 12y agoI presume it's been made with the author's own library: https://github.com/unconed/MathBox.js/ https://github.com/unconed/MathBox.js/
- ii 12y agoTo fold Julia fractal you only need to treat it like something simple and two-dimensional, project it onto a square grid, transform that grid, and then interpolate every cell using simple affine transformations. EDIT: Removed unneeded emotional junk EDIT 2: Affine transformations are just simple arithmetic operations between complex and dual numbers but nevertherless matrices are simpler!
- darsham 12y agoMaybe someone can help me understand this. On slide 39 of the first slideshow, it says : "For any irrational power p, there are an infinite number of solutions to z^p=c, all lying on a circle." This means that most of the solutions have an angle larger than a full circle, right ? But if complex numbers can be represented as the sum of the real and complex parts, how can their angle be superior to 360 degrees ?
- zygy 12y agoThis is the 'collapsing' that he talks about. If one of the complex solutions had a magnitude of 1 and an angle of 395 degrees, e.g., it would be equal to a number with a magnitude of 1 and an angle of 35 degrees.
- darsham 12y agoThanks (and thanks to the other replies !). So a complex number has a single representation when using real and imaginary parts, but an infinity of representations if you use angle and magnitude (just add or subtract 2π radians). I guess that should have been clear from the article, I just needed to sleep on it.
- NAFV_P 12y agoz = x + i*y = e^(a + i*b) = e^a*(e^(i*b)) = e^a(cos(b)+i*sin(b)) Both sin and cos are many to one functions. In the equation above, replacing b with b + 2*pi*t where t is any positive or negative integer, would result in the same complex number.
- NaNaN 12y agoWe've used the same number for those different rotations. That does not do anything directly about irrational power p. If p is a rational number represented by m/n, then 360 / (m/n) * m = 360n So there are multiples of 360 that are divisible by m/n, and we can get a unique complex number from z^p. Now p is irrational, what happens? According to what you said, there are so many different graphs that can not be merged into just one.
- NAFV_P 12y agoDare I mention hypercomplex numbers? The first ones you come across are quaternions, represented by set H. Something I learned a few years ago, in set C there are two solutions to x^2+1=0, but in set H the set of solutions forms a Lie group, or basically an infinite number of solutions.